Minimax Optimal Algorithms with Fixed-$k$-Nearest Neighbors
Abstract
This paper presents how to perform minimax optimal classification, regression, and density estimation based on fixed- nearest neighbor (NN) searches. We consider a distributed learning scenario, in which a massive dataset is split into smaller groups, where the -NNs are found for a query point with respect to each subset of data. We propose \emph{optimal} rules to aggregate the fixed--NN information for classification, regression, and density estimation that achieve minimax optimal rates for the respective problems. We show that the distributed algorithm with a fixed over a sufficiently large number of groups attains a minimax optimal error rate up to a multiplicative logarithmic factor under some regularity conditions. Roughly speaking, distributed -NN rules with groups has a performance comparable to the standard -NN rules even for fixed .
Cite
@article{arxiv.2202.02464,
title = {Minimax Optimal Algorithms with Fixed-$k$-Nearest Neighbors},
author = {J. Jon Ryu and Young-Han Kim},
journal= {arXiv preprint arXiv:2202.02464},
year = {2024}
}
Comments
65 pages, 5 figures. The manuscript has been revised from scratch compared to the previous version. Notable differences include (1) updated statements and corrected proofs for classification and regression, (2) explicit statements and proofs for distance-selective rules, and (3) new analogous estimators for density estimation